Here, E(x) is a function from the space of states to the real numbers; in physics applications, E(x) is interpreted as the energy of the configuration x. The parameter β is a free parameter; in physics, it is the inverse temperature. The normalizing constantZ(β) is the partition function. However, in infinite systems, the total energy is no longer a finite number and cannot be used in the traditional construction of the probability distribution of a canonical ensemble. Traditional approaches in statistical physics studied the limit of intensive properties as the size of a finite system approaches infinity (the thermodynamic limit). When the energy function can be written as a sum of terms that each involve only variables from a finite subsystem, the notion of a Gibbs measure provides an alternative approach. Gibbs measures were proposed by probability theorists such as Dobrushin, Lanford, and Ruelle and provided a framework to directly study infinite systems, instead of taking the limit of finite systems.

A measure is a Gibbs measure if the conditional probabilities it induces on each finite subsystem satisfy a consistency condition: if all degrees of freedom outside the finite subsystem are frozen, the canonical ensemble for the subsystem subject to these boundary conditions matches the probabilities in the Gibbs measure conditional on the frozen degrees of freedom.

The Gibbs measure of an infinite system is not necessarily unique, in contrast to the canonical ensemble of a finite system, which is unique. The existence of more than one Gibbs measures is associated with statistical phenomena such as symmetry breaking and phase coexistence.

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An example of the Markov property can be seen in the Gibbs measure of the Ising model. The probability for a given spin σk to be in state s could, in principle, depend on the states of all other spins in the system. Thus, we may write the probability as

.

However, in an Ising model with only finite-range interactions (for example, nearest-neighbor interactions), we actually have

,

where Nk is a neighborhood of the site k. That is, the probability at site k depends only on the spins in a finite neighborhood. This last equation is in the form of a local Markov property. Measures with this property are sometimes called Markov random fields. More strongly, the converse is also true: any positive probability distribution (nonzero density everywhere) having the Markov property can be represented as a Gibbs measure for an appropriate energy function.[1] This is the Hammersley–Clifford theorem.

Given a configuration ω ∈ Ω and a subset , the restriction of ω to Λ is . If and , then the configuration is the configuration whose restrictions to Λ1 and Λ2 are and , respectively. These will be used to define cylinder sets, below.

The set of all finite subsets of .

For each subset , is the σ-algebra generated by the family of functions , where . This σ-algebra is just the σ-algebra of cylinder sets on the lattice.

To help understand the above definitions, here are the corresponding quantities in the important example of the Ising model with nearest-neighbor interactions (coupling constant J) and a magnetic field (h), on Zd: